Problem 6

Consider an isosceles triangle. Let

R

be the radius of its circumcircle and

r

the radius of its inscribed circle. Prove that the distance

d

between the centers of these two circles is

\sqrt{R(R-2r)}

Step 5 of 6: Use power of I with the circumcircle

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P,Q=\text{intersections of }OI\text{ extended with the circumcircle}\quad\Longrightarrow\quad AI\cdot IL=PI\cdot QI

Detailed analysis

Extend $OI$ to meet the circumcircle at $P,Q$ . The two secants through $I$ give equal powers of $I$ : the secant $AIL$ gives $AI\cdot IL$ , while the diameter line gives $PI\cdot QI$ . Thus $AI\cdot IL=PI\cdot QI$ .